period surplus (A* _{j}*−

*φsR*

*−*

^{d}*w). Workers earn wagew(A*

*) if the bargaining succeeds and*

_{j}*w*otherwise. The entrepreneur earns operating profit

*A*

*−*

_{j}*w(A*

*) per match if the bargaining succeeds. Otherwise, she liquidates the associated physical capital, which depreciates by factor 0*

_{j}*< s <*1, and earns the deposit interest rate,

*R*

*, on the scrap value. Nash bargaining with worker’s bargaining power*

^{d}*η*∈(0,1) yields the wage

*w(A** _{j}*) =

*η(A*

*−*

_{j}*sφR*

*) + (1−*

^{d}*η)w.*(1.3) The negotiated wage exceeds the worker’s outside option if (A

*−*

_{j}*sφR*

*)*

^{d}*> w.*

Using wage (1.3) and the number of employed workers, operating profits read

Π(A_{j}*, n** _{j}*) =

*n*

*[(1−*

_{j}*η)(A*

*−*

_{j}*w) +ηsφR*

*], (1.4) are linear in firm size and using (1.1) rewrite*

^{d}Π(A_{j}*, K** _{j}*) =

*ρ(A*

*)·*

_{j}*K*

_{j}*,*with

*ρ(A*

*) = 1*

_{j}*φ*[(1−*η)(A** _{j}*−

*w) +ηsφR*

*]. (1.5)*

^{d}### 1.4.2 Financial frictions

Entrepreneurs can borrow*d**j* units of capital at net interest rate*R** ^{l}*, subject
to collateralized borrowing as in Kiyotaki and Moore (1997),

*R*^{l}*d**j*≤*λe**j**,* (1.6)

where *e** _{j}* denotes collateral and parameter

*λ*summarizes the quality of financial institutions, e.g. its ability to seize profits. Only funds invested in the firm serve as collateral. Assume

*A*

*j*is large enough to always ensure that the return on capital exceeds the borrowing rate,

*R*

^{l}*< ρ(A*

*).*

_{j}### 1.4.3 Investment problem

An entrepreneur’s state vector consists of her entrepreneurial ability, *A** _{j}*,
cash,

*x*

*, and the inherited firm’s current equity,*

_{j}*k*

*. The government levies tax rate*

_{j}*τ*

*on*

_{e}*k*

*and*

_{j}*τ*

*on*

_{s}*x*

*. The entrepreneur allocates cash and equity into a save deposit and/or invests in the firm. She can use funds invested in the firm as collateral to borrow*

_{j}*d*

*. Debt must be repaid at net interest*

_{j}rate*R*^{l}*> R** ^{d}*. Appendix 1.B proves that it is optimal to invest the complete
inheritance in the firm rather than deposit at the lower interest rate

*R*

*. It follows for funds invested by the heir:*

^{d}*e*

*= (1−*

_{j}*τ*

*)k*

_{e}*+ (1−*

_{j}*τ*

*)x*

_{s}*.*

_{j}^{12}Here, I anticipate this result for brevity. Denote the total capital input

*K*

*=*

_{j}*e*

*+*

_{j}*d*

*.*

_{j}The debt choice problem reads:

*V** _{e}*(x

_{j}*, k*

_{j}*, A*

*) max*

_{j}

_{d}

_{j}*ρ(A*

*)[(1−*

_{j}*τ*

*)x*

_{s}*+ (1−*

_{j}*τ*

*)k*

_{e}*+*

_{j}*d*

*]−*

_{j}*R*

^{l}*d*

*(1.7) s.t.*

_{j}*R*

^{l}*d*

*≤*

_{j}*λ[(1*−

*τ*

*)x*

_{s}*+ (1−*

_{j}*τ*

*)k*

_{e}*]*

_{j}*d**j* ≥0.

In the optimum, entrepreneurs exhaust the borrowing constraint, choosing maximum leverage,

*d** _{j}* =

*λ*

*R** ^{l}*[(1−

*τ*

*)x*

_{s}*+ (1−*

_{j}*τ*

*)k*

_{e}*], (1.8) which implies a total capital input of*

_{j}*K** _{j}*=

1 + *λ*
*R*^{l}

[(1−*τ** _{s}*)x

*+ (1−*

_{j}*τ*

*)k*

_{e}*]. (1.9) Linearity of operating profit and (1.8), are used to identify the value of a firm,*

_{j}*V** _{e}*(x

_{j}*, k*

_{j}*, A*

*) =*

_{j}*ρ(A*]

*)[(1−*

_{j}*τ*

*)x*

_{s}*+ (1−*

_{j}*τ*

*)k*

_{e}*], (1.10) where*

_{j}*ρ(A*]

*) =*

_{j}*ρ(A*

*)*

_{j}1 +_{R}^{λ}_{l}

−*λ*denotes the (leveraged) return on
equity-financed capital, which includes the exhaustion of the borrowing constraint
and subtraction of the costs of debt.

Given the Leontieffproduction function, the optimal number of employed workers is

*n** _{j}*=

*K*

_{j}*φ*= 1

*φ*

1 + *λ*
*R*^{l}

[(1−*τ** _{s}*)x

*+ (1−*

_{j}*τ*

*)k*

_{e}*]. (1.11) All*

_{j}*n*

*workers earn and consume wage income*

_{j}*w(A*

*).*

_{j}Children who have chosen to follow in their parents’ footsteps and manage the firm solve problem (1.7). They choose debt (1.8) and new employment

12The proof relies on the assumption*ρ(A** _{j}*)

*> R*

^{l}*> R*

*: as long as the firm is more profitable than saving, a risk-neutral agent will invest exclusively in the firm. As long as leverage has a positive net return,*

^{d}*ρ(A*

*)*

_{j}*> R*

*, the risk-neutral agent will also exhaust the borrowing constraint (see Appendix 1.B).*

^{l}(1.11). Denote their indirect utility from problem (1.7) by*V** _{e}*(x

_{j}*, k*

_{j}*, A*

*). As-sume that parents, who are the firm founders, invest equity*

_{j}*k*

*−1 and have entrepreneurial ability*

_{j}*A*

*−1. They solve the related problem and choose debt*

_{j}*d*

*−1=*

_{j}

_{R}

^{λ}*l*

*k*

*−1, implying total capital input of*

_{j}*K*

*−1= (1 +*

_{j}

_{R}

^{λ}*l*)k

*−1 and firm size*

_{j}*n*

*−1=*

_{j}*φ*

^{−}

^{1}(1 +

_{R}

^{λ}*l*)k

*−1in terms of employment.*

_{j}### 1.4.4 Career choice

Between the heir’s and parent’s investment problem, the heir makes a discrete
career choice: to either manage the family firm as an entrepreneur or sell
the firm and pursue her talents elsewhere, i.e. in a job outside the family
company.^{13}

The entrepreneurial path yields indirect utility*V** _{e}*(x

_{j}*, k*

_{j}*, A*

*), but the heir incurs idiosyncratic utility cost*

_{j}*drawn from a distribution with cumulative density function*

_{j}*F().*

*subsumes the costs of acquiring human capital to become a manager, working with parents in their firm, and talent for jobs outside of the family business.*

_{j}*can also be interpreted as impatience: an impatient heir might want to sell the company and consume now rather than manage and consume profits over time.*

_{j}The talent path yields an outside wage*w** _{out}* and non-stochastic interest

*R*

*on the cash inheritance,*

^{d}*x*

*, plus the scrap value of equity,*

_{j}*sk*

*. Both, cash and sold capital, are then subject to the non-corporate tax rate,*

_{j}*τ*

*. The children’s career choice has external effects on the workers’ human capital:*

_{s}if the firm is scrapped, the*n** _{j}*−1 workers formerly employed by the parents
lose their high-tenure jobs and will earn and consume the outside wage.

The difference*w(A** _{j}*)−

*w >*0 reflects the substantial earnings losses after lay-off. The children’s career choice is the main motivation for optimal tax deductions: an heir may abstain from pursuing her own talents if the tax penalty on this path is high enough.

Formally, the heir chooses the entrepreneurial path if the value of

en-13The heir’s outside wage need not equal the workers’ outside wage. Assuming that an entrepreneur’s child has benefited from excellent education, she can be a “Jack of all trades”

(Lazear, 2005) with the human capital to earn a higher wage than the attached workers
outside of the company (w* _{out}*≥w).

trepreneurship net of utility costs exceeds the utility of firm liquidation,
*V** _{e}*(x

_{j}*, k*

_{j}*, A*

*)−≥*

_{j}*w*

*+ (1−*

_{out}*τ*

*)R*

_{s}*[x*

^{d}*+*

_{j}*sk*

*]. (1.12) The solution to the discrete choice problem is a reservation talent,*

_{j}¯

*(x*_{j}*, k**j**, A**j*)≡*V**e*(x_{j}*, k**j**, A**j*)−*w**out*−(1−*τ**s*)R* ^{d}*[x

*+*

_{j}*sk*

*j*], (1.13) and the firm is liquidated if

_{j}*>(x*¯

_{j}*, k*

_{j}*, A*

*).Denote the indirect utility of the career choice problem (1.12) by*

_{j}*V*(x

_{j}*, k*

*j*

*, A*

*j*).

### 1.4.5 Bequest choice

Assume a bequest policy function:^{14} the parent will always bequeath the
company’s equity *k** _{j}* =

*k*

*−1 and a cash bequest,*

_{j}*x*

*, which is an increasing function of*

_{j}*k*

*j*−1 as a proxy of wealth,

*x** _{j}*= ˜

*x*

*(k*

_{j}*−1*

_{j}*, τ*

*), with ˜*

_{s}*e*

_{x}

_{j}

_{,k}*≥0, and ˜*

_{j−1}*e*

_{x}

_{j}

_{,τ}*≤0.*

_{s}*e*˜_{x}_{j}_{,k}_{j}_{−}_{1} denotes the elasticity of cash bequests to capital bequests and ˜*e*_{x}_{j}_{,τ}* _{s}*
denotes the tax elasticity of cash bequests. For tractability and to rule out tax
sheltering, assume ˜

*e*

*x*

_{j}*,τ*

*= 0, i.e. parents who are entrepreneurs do not adjust the cash bequest in response to changes in the tax rate of the continued firm.*

_{e}Excess profitability of equity (Appendix 1.C) yields the choice*k** _{j}* =

*k*

*−1, which is credible in light of low empirical estimates of the tax bequest elas-ticity and special role of a family business for donors: Slemrod and Kopczuk (2001) identify a long-run tax elasticity of between -0.1 and -0.16. The tax elasticity of business assets is probably smaller: first, entrepreneurs, espe-cially firm founders, tend to take pride in their work and their business and feel some responsibility for their workers (Kammerlander, 2016). Secondly, a company is not easily divided into small chunks which are consumed or bequeathed without friction.*

_{j}In my framework, tax sheltering is ruled out by assumption, but as shown by Alstadsaeter et al. (2014), favourable taxation of business assets leads

14See Appendix 1.C for a microfoundation of policy functions. The main text deviates
from the microfoundation by assuming ˜*e*_{x}_{j}_{,τ}* _{e}*= 0.

to a shift in portfolio choices. In Germany, §13a spawned *Cash-GmbHs,*
pseudo-companies designed to hoard liquid assets for the next generation.

### 1.4.6 Taxation and divestment

This subsection provides the first motivation for a deduction of business assets. Taxation might lead to divestment and lay-offs in a continued firm.

When employment and capital are complements, employment rises with
capital input. Taxation can reduce employment via four channels: (i) if the
elasticity of bequests with respect to taxes ˜*e*_{x}_{j}_{,τ}* _{s}* is negative, taxes reduce
bequests; (ii) taxation reduces capital by reducing the net value of cash or
capital the heir receives; (iii) by reducing an heir’s asset holdings, the
bor-rowing constraint tightens; (iv) taxation affects career choice. The influence
of taxation on career choice (iv) is discussed in the presentation of the central
planner’s problem below. For now, focus on the bequest decision of parents
and the investment decisions of children. The effects of inheritance taxation
on employment are

*∂n*_{j}

*∂τ** _{s}* =−

*x*

*j*

*φ*

^{−}

^{1}

1 + *λ*
*R*^{l}

"

1 +1−*τ*_{s}*τ*_{s}*e*˜*x*_{j}*,τ*_{s}

#

*<*−*x**j**φ*^{−}^{1}

"

1 +1−*τ*_{s}*τ*_{s}*e*˜*x*_{j}*,τ*_{s}

# and

*∂n*_{j}

*∂τ**e*

=−*k** _{j}*−1

*φ*

^{−}

^{1}

1 + *λ*
*R*^{l}

*<*−*k** _{j}*−1

*φ*

^{−}

^{1}

*.*

The term−*x*_{j}*φ*^{−}^{1}h

1 +^{1}^{−}_{τ}^{τ}^{s}

*s* *e*˜_{x}_{j}_{,τ}* _{s}*i

is the direct effect of taxation on employment
if entrepreneurs cannot borrow (λ= 0). It is the sum of channels (i) and
(ii). Each unit of net cash bequest invested into the company allows for the
employment of*φ*^{−}^{1} additional workers. Similarly, −*k** _{j}*−1

*φ*

^{−}

^{1}is the direct effect of taxation of corporate assets absent debt (λ= 0).

Additionally, a lower net value of cash or equity tightens the borrowing
constraint (iii). With each unit of cash or equity the heir can increase debt,
1 +_{R}^{λ}_{l}

*>*1. Leverage increases total capital input and amplifies the
detri-mental effect of taxation on employment. When we study optimal taxation
with entrepreneurship but ignore borrowing constraints, we overlook the
amplifying effect leverage can have on labour demand.

Importantly, a reduction of employment may translate into earnings losses for workers because the wage paid in the firm exceeds the workers’ outside

option. This channel is widely overlooked in the optimal taxation literature based on neoclassical labour markets. In a neoclassical labour market, tax-ation reduces aggregate capital, which in turn depresses wages via general equilibrium effects. In this model, taxation can reduce the wages for some agents considerably, if company heirs are forced to lay offworkers.

If an inheritance is not accompanied with sufficient liquid assets to pay the tax liability, heirs must reduce the firm size and lay offworkers to finance the taxes, i.e. heirs operate firms with smaller total capital and employment if

*τ*_{s}*x** _{j}*+

*τ*

_{e}*k*

*−1*

_{j}*< x*

_{j}*.*

Empirically, Germany’s estate and gift tax statistic lets us compute the ratio
of liquid assets to company assets. As an example, consider an heir who
inherits some cash and a company worthe4m (after deduction ofe400,000,
see Table 1.1). The applicable tax rates for this inheritance are *τ** _{s}* = 19%

and*τ**e* = (1−0.4)×19% = 11.4% for business assets in the year 2002 (see
Table 1.2). Table 1.3 shows that the average cash bequest this heir can expect
is worth 27% of the firm’s value. It follows 19%·27% + 11.4%*<*27% and the
average heir can pay the inheritance tax liabilities using only the transferred
cash. In fact, even if there were no favourable treatment of company assets
(τ* _{e}* =

*τ*

*), the average company heir could pay the tax liabilities out of the parent’s pocket.*

_{s}Note that recipients of company assets have access to deferments. The German government grants a 10 year window over which bequest taxes can be deferred free of interest for business heirs. Additional interest-bearing deferment is possible if paying taxes is classified an undue hardship for the beneficiary. This has two consequences: as Gale and Slemrod (2001) note, the present value of the tax is lower because of intertemporal discounting.

Secondly, not all taxes need to be paid out of cash or by selling the company.

Instead the government acts as an extended financial sector, providing loans
free of collateral constraints. Those findings are in line with Gale and
Slemrod (2001) who point out that the accumulation of assets to finance
an expected tax is good business practice. Holtz-Eakin et al. (2001) find
that up to 58% of business owners can pay their estate taxes using only
liquid assets. Deferments,*inter vivo*gifts, life insurance, cash transfers, and

financial markets cast doubt on the idea that taxation leads to divestment as outlined in this subsection. How, then, can we justify the deductions granted by lawmakers? The main rationale in this paper is the externality of an heir’s career choice discussed in the next section.